Section outline

  • Collapse Expand

    MM7052 Topology

    Collapse all Expand all

    Course content:

    The course treats the foundations of general set topology (topological spaces, continuity, compactness, connectedness, quotient topologies), the fundamental group and the classification of closed surfaces.

    Teachers

    Lectures: Gregory Arone  
     E-mail: gregory dot arone at math dot su dot se

    Exercise sessions: Coline Emprin  
     E-mail: Coline dot Emprin at math dot su dot se 

    Course literature

    John M. Lee, Introduction to Topological Manifolds, 2nd edition. (Springer)

    As extra literature one may use J.R. Munkres, Topology (Prentice Hall) and A. Hatcher, Algebraic topology (freely available online).

    Schedule in TimeEdit

    We meet: Mondays 13:00 - 16:00
    Except for the first and last lecture, the time will be divided between:
    Exercises: 13:00 - 14:00.
    and
    Lectures: 14:00 - 16:00

    Note that the lecture-plan below is tentative and might change slightly.

    Howework assignments

    A homework assignment will be posted once in two weeks Monday, with deadline Thursday the following week. The homework assignment is discussed at the exercise session after the deadline. The homework gives up to 3 bonus points on the exam (and on the following re-exam).

    You are allowed to collaborate on the homework assignments, and you also are allowed to ask for help from the teachers, after making an effort to do them yourself. But you should write the assignments individually and not copy. Please do not seek help with the assignments on the internet and do not use ChatGPT or similar, not least because it often gives a wrong answer. The main purpose of the assignment is to help you master the material. The bonus points are just a bit of extra motivation. If you cheat, you will be cheating yourself.

    Written exam: December 15 (2025), 8:00 - 13:00 (place tba).
    Do not forget to register for the exam (at the latest 10 days before the exam)!

    The exam (and re-exam) will consist of 5 problems giving at most 30 points. 

    The grading for the written exam: E 16, D 19, C 22, B 25, A 28. (may be adjusted slightly)

    One of the  following five problems will appear on the exam (and re-exam)
    (I will roll a dice to determine which):

    1. Closed map lemma
    First prove Proposition 4.36a and 4.36b. Then prove Lemma 4.50.

    2. The fundamental group is a homotopy invariant
    Prove first Lemma 7.45 then Theorem 7.40.

    3. Attaching a disc
    Prove Proposition 10.13.

    4. Existence of the Universal Covering Space
    Show Step 1, 3 and 4 of Theorem 11.43.

    5. Classification of coverings
    Prove Theorem 12.18, but assume it to be known that the map
    "hat q", defined in the proof, is a covering map.

    In the proofs you are allowed to use any result of the book
    that appears earlier, as long as you write out the statement
    of the result you are using (for instance, if you want to
    use Proposition 2.15 then you write "We know from earlier
    that: A map between topological spaces is continuous if and
    only if the preimage of every closed subset is closed.").

    Note that the proofs of 3,4,5 do not need to be as detailed as in the
    book, but need to contain an explanation of all steps.

    Note also that there is an errata with a (for instance) small correction to the
    proof of 3.

    Written  re-exam: tba.
    Do not forget to register for the exam (at the latest 10 days before the exam)!

    Exam returns take place during the student affairs office's opening hours Tuesdays 11:45-12:45.
    Before you come and pick up your exam, you need to fill in this form: https://survey.su.se/Survey/51990/en. When your exam is ready for return, you will receive an email from tentataerlamning@math.su.se.

    Examination Rules at the Department of Mathematics


    Resources
  • Collapse Expand

    Lecture 1

    Topological spaces, convergence and continuity, homeomorphisms, closed set, closure, interior, limit point, Hausdorff spaces.

    Reading: Chapter 2 (pages 19-35).

    Some recommended
    Exercises: 2.9, 2.10, 2.14, 2.21, 2.28, 2.40, 2.42.
    Problems: 2-7, 2-14, 2-15

  • Collapse Expand

    Lecture 2

    Bases, countability axioms, separable spaces. Subspace topology.

    Reading: Chapter 2 (cont.) and beginning of Chapter 3 (pages 36-53).

    Some recommended
    Exercises: 2.51, 2.54, 3.1, 3.3, 3.6, 3.12.
    Problems: 2-21, 2-24.

  • Collapse Expand

    Lecture 3

    Open and closed maps, embeddings, product spaces, topological groups. Manifolds.

    Reading: Chapter 3 (pages 54-64 and 77-78)

    Some recommended
    Exercises: 3.13, 3.26, 3.29, 3.32, 3.38, 3.83.
    Problems: 3-7, 3-20.

  • Collapse Expand

    Lecture 4

    Disjoint unions, quotient spaces, adjunction spaces, orbit spaces (cursory: proof of Thm 3.79).

    Reading: Chapter 3 (pages 64-77 and 78-81)

    Some recommended
    Exercises: 3.40, 3.43, 3.46, 3.61, 3.63, 3.72.
    Problems: 3-14, 3-15, 3-16.

  • Collapse Expand

    Lecture 5

    Connectedness and path-connectedness.

    Reading: Chapter 4 (pages 85-93)

    Some recommended
    Exercises: 4.4, 4.5, 4.10, 4.14, 4.22.
    Problems: 4-2, 4-4, 4-8.

  • Collapse Expand

    Lecture 6

    Compactness, local compactness, proper maps (cursory: paracompactness, normal spaces and partitions of unity, proof of Lemma 4.72).

    Reading: Chapter 4 (pages 94-121).

    Some recommended
    Exercises: 4.29, 4.38, 4.49, 4.67, 4.78, 4.79.
    Problems: 4-18, 4-20.

  • Collapse Expand

    Lecture 7

    The fundamental group.

    Reading: Chapter 7 (pages 183-200).

    Some recommended
    Exercises: 7.8, 7.15, 7.23.
    Problems: 7-2, 7-5, 7-10. 7-11.


  • Collapse Expand

    Lecture 8

    Homotopy equivalence. Homotopy invariance of the fundamental group. Categories and functors

    Reading: Chapter 7 (pages 200-214). Higher homotopy groups not mentioned in class

    Some recommended
    Exercises: 7.35, 7.42, 7.58.
    Problems: 7-3, 7-8, 7-16.
  • Collapse Expand

    Lecture 9

    Fundamental group of the circle.

    Reading: Chapter 8.

    Some recommended
    Exercises: 8.7.
    Problems: 8-1, 8-2, 8-4, 8-6.

  • Collapse Expand

    Lecture 10

    Covering maps. Fundamental groups of orbit spaces.

    We will cover parts of Chapter 11 and 12, using covering maps as a tool to calculate fundamental groups.
    The following lecture will cover the remaining parts.


    Reading: Chapter 11 (pages 277-293). Proof of Theorem 11.18 and pages 288-293 cursory. Chapter 12 (pages 308-314).

    Some recommended
    Exercises: 11.1, 11.9
    Problems: 11-1, 11-2, 11-4.

  • Collapse Expand

    Lecture 11

    The universal cover. Classification of coverings.

    Reading: Chapter 11 (pages 294-302). Chapter 12 (pages 315-318).

    Some recommended
    Problems: 11-18, 11-19, 12-7.
  • Collapse Expand

    Lecture 12

    Free groups, amalgamated free products, statement of the Seifert-Van Kampen theorem and examples.

    Reading: Chapter 9 (pages 233-244). Chapter 10 (pages 251-264).
    For CW-complexes you only need to know their definition (pages 129-132)..

    Some recommended
    Exercises: 9.10, 10.9.
    Problems: 9-2, 9-4, 10-5, 10-6.
  • Collapse Expand

    Lecture 13

    Proof of the Seifert-Van Kampen theorem.
    Polygonal presentations of surfaces. Fundamental groups of surfaces and their abelianizations.

    Reading: Chapter 6 (pages 159-172). Chapter 10 (pages 264-273).

    Some recommended
    Exercises: 6.11, 10.18, 10.20.
    Problems: 6-3, 6-6 (skip question about Euler char.), 10-7, 10-18.

  • Collapse Expand

    Lecture 14

    End of proof of the Seifert - van Kampen theorem. A survey of surfaces.

    Reading: Chapter 5, p.147-155, Chapter 6, p.173-178.

    Some recommended
    Exercises: 5.31, 5.34, 5.40.
    Problems: 5-17.

    No more assignments will be posted.

  • Collapse Expand

    Lecture 15

    This will be a review. We will go over various problems, and you will have a chance to ask questions.